As promised, I am going to present small part of my mathematical research in my blog and interested people can just download the paper for more detail. Often case, I just write down the abstract that is already part of my paper. I also will write down some history of how things were developed, which is otherwise not written in the paper or article.
I started a few months ago in the summer of 2014. Obviously, the $3n+1$ problem (or the Collatz conjecture) is not a field I have worked with before 2014. I do mostly algebraic geometry and commutative algebra. So I wanted something new. I personally thought that it would be easier to work with binary representations of the Collatz sequence and it turned out that, at least for me, I could understand the sequence better that way. I set myself that goal of looking for something new for at least a year until if things were not anymore promising I would just call an end to this research and start something else. Well at least I think I did find something new. Far from any form of solution to the conjecture or even something that might prove significant for the research community. It’s only significance is probably that it is a very easy to understand characterisation (or suficiency) for the Collatz conjecture to be true. Probably, after this or at least after I get this somehow published, I will not be doing much more and try to look at other things and then maybe every now and then take a glance at the Collatz conjecture again. I always need some change whenever I do something for a long time. If I do get lucky I might see something again, but there are no gaurantees. Well let me show the abstract of the paper:
Here we investigate the odd numbers in Collatz sequences (sequences arising from the $3n+1$ problem). We are especially interested in methods in binary number representations of the numbers in the sequence. In the first section, we show some results for odd Collatz sequences using mostly binary arithmetics. We see how some results become more obvious in binary arithmetic than in usual method of computing the Collatz sequence. In the second section of this paper we deal with some known results and show how we can use binary representation and OCS from the first section to prove some known results. We give a generalization of a result by Andaloro [1] and show a generalized sufficient condition for the Collatz conjecture to be true: If for a fixed natural number $n$, the Collatz conjecture holds for numbers congruent to $1$ modulo $2^n$, then the Collatz conjecture is true.
The paper thus provides a sequence of sufficiency set whose set-theoretic limit is the set $\{1\}$. Similar sequence of sufficiency set has been found before (the natural density approaches $0$ but the set-theoretic limit is not necessary the singleton containing $1$). I tend to think that this one is the simplest one. The paper in preprint form can be found here!
[1] P. Andaloro, On Total Stopping Times under $3x+1$ Iteration, Fibonacci Quarterly 2000, Vol. 38, No. 1, p. 73-78